ISC (Science) ISC Class 12 - CISCE Question Bank Solutions

Statement 1: The intersection of two equivalence relations is always an equivalence relation.

Statement 2: The Union of two equivalence relations is always an equivalence relation.

Which one of the following is correct?

If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.

Given an example of

a row matrix which is also a column matrix,

If A = `[[3,1] , [7,5]]`, find the values of x and y such that A² + xI₂ = yA.

If cos^-1 x + cos ^-1 y + cos ^-1 z = π , prove that x² + y² + z² + 2xyz = 1.

If y = `(x sin^-1 x)/sqrt(1 -x^2)`, prove that: `(1 - x^2)dy/dx = x + y/x`

If `tan^-1 ((x - 1)/(x + 1)) + tan^-1 ((2x - 1)/(2x + 1)) = tan^-1 (23/36)` = then prove that 24x² – 23x – 12 = 0

The value of cosec `[sin^-1((-1)/2)] - sec[cos^-1((-1)/2)]` is equal to ______.

Solve for x: `sin^-1(x/2) + cos^-1x = π/6`

If sin^–1x + sin^–1y + sin^–1z = π, show that `x^2 - y^2 - z^2 + 2yzsqrt(1 - x^2) = 0`

Solve:

sin^–1 (x) + sin^–1 (1 – x) = cos^–1 x

Given two matrices A and B 

`A = [(1,-2,3),(1,4,1),(1,-3, 2)]  and B  = [(11,-5,-14),(-1, -1,2),(-7,1,6)]`

find AB and use this result to solve the following system of equations:

x - 2y + 3z = 6, x + 4x + z = 12, x - 3y + 2z = 1

Show that (A + A') is symmetric matrix, if `A = ((2,4),(3,5))`

If A is a square matrix of order 3, then |2A| is equal to ______.

Assertion: Let the matrices A = `((-3, 2),(-5, 4))` and B = `((4, -2),(5, -3))` be such that A¹⁰⁰B = BA¹⁰⁰

Reason: AB = BA implies AB = BA for all positive integers n.

A matrix which is both symmetric and skew symmetric matrix is a ______.

If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`

Let \[f\left(x\right) = x^3\] be a function with domain {0, 1, 2, 3}. Then domain of \[f^{-1}\] is ______.

Let A = R – {2} and B = R – {1}. If f: A `→` B is a function defined by f(x) = `(x - 1)/(x - 2)` then show that f is a one-one and an onto function.

Which one of the following graphs is a function of x?

Graph A	Graph B